Olympiad problems

2008 Estonia Team Selection Test, 3

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2018 PUMaC Individual Finals A, 2

Tags: function

Find all functions $f:\mathbb{R^{+}}\to\mathbb{R^+}$ such that for all $x,y\in\mathbb{R^+}$ it holds that $$f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).$$

1996 Bosnia and Herzegovina Team Selection Test, 2

Tags: number theory , function , euler , eulers function , greatest common divisor

$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function $b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$

2007 Serbia National Math Olympiad, 1

Tags: induction , function , algebra

Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

2012 AMC 12/AHSME, 25

Tags: function

Let $f(x)=|2\{x\} -1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation $$nf(xf(x)) = x$$ has at least $2012$ real solutions $x$. What is $n$? $\textbf{Note:}$ the fractional part of $x$ is a real number $y= \{x\}$, such that $ 0 \le y < 1$ and $x-y$ is an integer. $ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\ 64 $

2002 Iran Team Selection Test, 10

Tags: matrix , function , vector , geometry , abstract algebra , rectangle , linear algebra

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

2011 Czech and Slovak Olympiad III A, 6

Tags: functional equation , function , algebra

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$, we have \[ f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.\]

2007 Bulgaria National Olympiad, 3

Tags: polynomial , algebra , search , function , trigonometry

Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$. [i]O. Mushkarov, N. Nikolov[/i] [hide]No-one in the competition scored more than 2 points[/hide]

2023 Belarus - Iran Friendly Competition, 2

Tags: polynomial , function , algebra

Find all strictly monotone functions $f : \mathbb{R} \to \mathbb{R}$ such that some polynomial $P(x, y)$ satisfies the equality $$f(x + y) = P(f(x), f(y))$$ for all real numbers $x$ and $y$

2011 Romanian Masters In Mathematics, 2

Tags: perpendicular bisector , geometry , geometric transformation , vector , function , combinatorial geometry , reflection

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

1981 AMC 12/AHSME, 17

Tags: function , domain , algebra

The function $f$ is not defined for $x=0$, but, for all non-zero real numbers $x$, $f(x)+2f\left( \frac1x \right)=3x$. The equation $f(x)=f(-x)$ is satisfied by $\text{(A)} ~\text{exactly one real number}$ $\text{(B)}~\text{exactly two real numbers}$ $\text{(C)} ~\text{no real numbers}$ $\text{(D)} ~\text{infinitely many, but not all, non-zero real numbers}$ $\text{(E)} ~\text{all non-zero real numbers}$

2022 Moldova Team Selection Test, 5

Tags: algebra , function , functional equation

The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies: $1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$ $2) f(20^{22})=f(22^{20});$ $3) f(2021)=2022$. Find all possible values of $f(2022)$.

2008 Brazil National Olympiad, 2

Tags: algorithm , function , number theory

Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.

1998 Korea - Final Round, 3

Tags: probability , function , combinatorics

Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions: 1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$ 2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$ Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.

2014 China Team Selection Test, 2

Tags: induction , function , relatively prime , number theory

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

1983 Miklós Schweitzer, 10

Tags: geometry , algebra , function , domain , advanced fields

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

2003 China Team Selection Test, 3

Tags: combinatorics , function

Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.

1985 National High School Mathematics League, 3

Tags: function

If $\arccos\frac{4}{5}-\arccos\left(-\frac{4}{5}\right)=\arcsin x$, then $\text{(A)}x=\frac{24}{25}\qquad\text{(B)}x=-\frac{24}{25}\qquad\text{(C)}x=0\qquad\text{(D)}$ No such $x$

2013 ELMO Shortlist, 8

Tags: number theory , modular arithmetic , algebra , counting , function , distinguishability , polynomial

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

1991 Putnam, B2

Tags: algebra , superior algebra , function , functional equation

Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

2009 All-Russian Olympiad, 3

Tags: function , algebra , trigonometry

How many times changes the sign of the function \[ f(x)\equal{}\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\] at the interval $ \left[0, \frac{2009\pi}{2}\right]$?

PEN P Problems, 12

Tags: calculus , integration , inequalities , floor function , function , logarithm

The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

1987 Traian Lălescu, 2.2

Tags: calculus , integration , inequalities , function

Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that $$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$ and find the circumstances under which equality happens.

1977 Polish MO Finals, 1

Tags: algebra , limit , function

A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.

2010 AMC 12/AHSME, 25

Tags: function

For every integer $ n\ge 2$, let $ \text{pow}(n)$ be the largest power of the largest prime that divides $ n$. For example $ \text{pow}(144)\equal{}\text{pow}(2^4\cdot 3^2)\equal{}3^2$. What is the largest integer $ m$ such that $ 2010^m$ divides \[ \prod_{n\equal{}2}^{5300}\text{pow}(n)\text{?}\] $ \textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78$